The reliability of a 6-metre long simply supported uniform beam under a uniformly distributed load, w, is to be investigated. The beam can either fail in bending, shear or combined bending and shear. The random variables are w, the bending capacity M and the shear capacity V of the beam. The statistics of the random variables and the performance functions (failure means value of function is less than zero) of the three failure modes are as follows: Variable Mean C.O.V. M V 600 kNm 0.1 400 kN 0.15 Distribution Normal or lognormal Normal or lognormal W 100 kN/m 0.25 Normal or lognormal 91(x) = g(M,w) = M 0.125wL 92(x)=92(v, w) = V - 0.5wL WL WM 93(X)=93(M,V,w) = 1 - 8M 2V2 Perform Direct Monte Carlo simulation assuming that M, V and w are independent normal random variables, for N of (i) 100 and (ii) 1000. Remember to freeze the generated data. Estimate (a) the failure probability of each mode; (b) the failure probability of the beam; and (c) the Chi-square Goodness of Fit test at 95% confidence level for the generated data of the random variables. Repeat for the case where M, V and w are independent lognormal random variates.